Near-barrier fusion, breakup and scattering for the 9Be + 144Sm system

Nuclear Physics A 828 (2009) 233–252 www.elsevier.com/locate/nuclphysa

Near-barrier fusion, breakup and scattering for the 9Be + 144Sm system P.R.S. Gomes a , J. Lubian a,∗ , B. Paes a , V.N. Garcia a , D.S. Monteiro a , I. Padrón b , J.M. Figueira c , A. Arazi c , O.A. Capurro c , L. Fimiani c , A.E. Negri c , G.V. Martí c , J.O. Fernández Niello c,d , A. Gómez-Camacho e , L.F. Canto f a Instituto de Física, Universidade Federal Fluminense, Av. Litoranea s/n, Gragoatá, Niterói, R.J., 24210-340, Brazil b Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear, Playa, Ciudad de la Habana, Cuba c Laboratorio Tandar, Departamento de Física, Comisión Nacional de Energía Atómica,

Avda. General Paz 1499 B1650KNA, Buenos Aires, Argentina d Escuela de Ciencia y Tecnogía, Universidad Nacional de San Martí, Argentina e Departamento del Acelerador, Instituto Nacional de Investigaciones Nucleares,

Apartado Postal 18-1027, C.P. 11801, Mexico, D.F., Mexico f Instituto de Física, Universidade Federal do Rio de Janeiro, C.P. 68528, 21941-972 Rio de Janeiro, Brazil

Received 30 April 2009; received in revised form 7 July 2009; accepted 13 July 2009 Available online 18 July 2009

Abstract Fusion, breakup and scattering for the 9 Be + 144 Sm system at near barrier energies are investigated by different approaches. We show that at energies above the barrier there is a small complete fusion suppression when compared with predictions from a double folding potential and with a similar tightly bound system. At sub-barrier energies there is no significant deviation from the predictions using coupled channel calculations that do not include the breakup channel. The energy dependence of the optical potential does not show the usual threshold anomaly found in tightly bound systems. From a simultaneous analysis of fusion and scattering data we estimate the distance where breakup starts to occur. © 2009 Elsevier B.V. All rights reserved. PACS: 25.60.Pj; 25.60.Gc

* Corresponding author. Tel.: +55 21 2629 5815; fax: +55 21 2629 5887.

E-mail address: [email protected] (J. Lubian). 0375-9474/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2009.07.008

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Keywords: N UCLEAR REACTIONS 144 Sm(9 Be, X), E = 30–48 MeV; measured fusion and elastic σ , σ (θ), σ (E). 144 Sm(9 Be, X), (16 O, X), E = 30–48 MeV; analyzed fusion σ , breakup mechanism; deduced universal fusion function. Delayed X-ray technique. Coupled channel calculations

1. Introduction In the last years, the influence of the breakup of weakly bound nuclei on elastic scattering and on the fusion processes has been widely studied [1]. There are different approaches for these studies, and usually one finds works in the literature that use one or a limited number of such approaches. Following the projectile breakup, several processes may occur: one of the fragments may fuse with the target in a process named incomplete fusion (ICF) or all the fragments may fly away from the target in a process called non-capture breakup (NCBU) [1]. Fusion of all the fragments with the target would lead to a sequential complete fusion (SCF). This process is expected to be less important than the direct complete fusion (DCF), where the whole projectile fuses with the target in a single step. The sum of these two processes is called complete fusion (CF). The inclusive fusion cross section arising from any of theses mechanisms is know as total fusion (TF) cross section. That is, σTF ≡ σICF + σSCF + σDCF . One of the most widely used approaches to investigate the influence of low breakup threshold energies on the fusion process is the measurement of total fusion excitation function of the system and the comparison of data with theoretical predictions of coupled channel calculations that do not take into account the breakup channel [2–6], or with predictions of complete CDCC calculations [13], which include the breakup channel. A few experiments report exclusive measurements of CF and ICF cross sections (see for example Refs. [2–6,15]). In such cases, CF is defined experimentally as the fusion of the total charge of the projectile with the target. In some experiments where the cross sections are measured with great accuracy, it is also possible to evaluate fusion barrier distributions [2–4]. These distributions give important information on the bare potential and on the specific channels influencing the fusion cross section at near-barrier energies. The low breakup threshold of the weakly bound systems studied in the present work influences the fusion cross section in two ways. First, through a static effect associated with the height and curvature of the Coulomb barrier. Second, through dynamical coupled-channel effects, resulting from the strong coupling with the breakup channel. Usually, only the combined static plus dynamic breakup effects on fusion of the weakly bound nuclei are investigated. However, very recently, a method to disentangle static and dynamic effects has been reported [16,17]. Another approach to address these problems is to investigate the energy dependence of the interaction nuclear potential [18–21], which is determined through fits of elastic scattering angular distributions. For tightly bound systems, the energy dependence of the interaction potential at energies close to the Coulomb barrier shows the usual threshold anomaly (TA) behavior [22,23], whereas for weakly bound systems the TA is often absent. Furthermore, for some weakly bound systems a TA with opposite behavior has been observed: the imaginary potential increases as the energy drops below the Coulomb barrier. This effect has been called breakup threshold anomaly (BTA) [24,25]. Depending on the net polarization potential being attractive or repulsive, fusion cross sections may be enhanced or suppressed at near barrier energies, when compared with one-dimensional barrier penetration models (or no-coupled calculations).

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The simultaneous analysis of fusion and elastic scattering (and corresponding total reaction cross section) can also be done, by performing optical model calculations. In these analyses, the interaction potential is split into a volume part, which is responsible for fusion reactions, and a surface part which accounts for direct reactions. This method allows the determination of the position where the breakup starts to take place by imposing the physical condition for the surface part of the potential to be outside the volume one [21]. In this paper we analyze a comprehensive set of data for the 9 Be + 144 Sm system at near barrier energies using different approaches, and compare the conclusions. We also compare our results with those for the tightly bound 16 O + 144 Sm system. Throughout this paper, we adopt the parameter-free double folding São Paulo potential (SPP) [26–28] as the bare interaction. Most of the data analyzed here have already been published [5,6]. However we present also some original experimental results, as elastic scattering data (including points at sub-barrier energies), the derived reaction and NCBU cross sections, and also the distance where breakup begins to occur. In Section 2 we present the experimental details, the quantities measured and derived from data for this system. In Section 3 we make the comparison of the complete and total fusion cross section data with theoretical predictions from coupled channel calculations including and not including the resonant states of 9 Be nucleus. In Section 4 we present a method which enables us to compare the fusion cross section data for the 9 Be + 144 Sm system with those of the tightly bound 16 O + 144 Sm system. In Section 5 we derive the cross sections for the direct (breakup) reactions. In Section 6 we show the investigation of the energy dependence of the interaction potential. In Section 7 we investigate the energy dependence of the fusion and direct reaction parts of the optical potential and we estimate the distance where the breakup process starts to occur by the simultaneous fits of fusion and elastic scattering data. Finally, in Section 8 we present a summary and conclusions. 2. Experimental details and quantities measured or derived from the data The experiments were performed at the Tandar Laboratory, Buenos Aires. For the fusion cross section measurements, 9 Be beams for ten energies in the range 30 MeV  Elab  44 MeV (The Coulomb barrier is around E = 33 MeV) were used. Angular distributions for elastic and inelastic scattering, for six energies from 33 MeV  Elab  41 MeV were also measured. Those data have been previously reported [5,6]. In addition, we report in the present paper original data for the elastic scattering at four bombarding energies, two at sub-barrier (Elab = 30.0 and 31.5 MeV) and two at above-barrier (Elab = 44 and 48 MeV) energies. Five 88.6% enriched 144 Sm targets, with thicknesses from 170 µg/cm2 to 220 µg/cm2 on carbon backings of 20 µg/cm2 were used. Complete fusion (CF) and incomplete fusion (ICF) cross sections were separately measured [5,6] by the delayed X-ray technique [7]. This is possible because the residue nuclei of both processes decay by electron capture with half lives from some minutes to many hours, including at least two generations of decay. This method has been applied in the past for tightly bound systems [8–11] and more recently [12] for the 6 He + 64 Zn system. For the 9 Be + 144 Sm system, the compound nuclei for the CF (153 Dy) decay mainly by neutron evaporation, 2n (151 Dy) and 3n (150 Dy) being dominant in our energy range, from 0.8 to 1.4 times the Coulomb barrier. For ICF, the main evaporation products are from the 1n evaporation channel (147 Gd with T1/2 = 38.1 hours). Other possible channels are 148 Gd (0n channel with T1/2 = 98 years) and 146 Gd (2n channel with T1/2 = 48.3 days), that could not be measured due to

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Fig. 1. Typical spectrum for the 9 Be + 144 Sm scattering for θlab = 64 degrees and Elab = 44.0 MeV.

their very long half lives. Consequently, only a lower limit of the ICF cross section could be determined. An aluminum catcher foil was placed a few mm behind the target. Its thickness of 1.6 mg/cm2 was chosen, such that it should stop the fusion residues but let the elastically scattered beam particles pass through. One surface barrier silicon detector was placed at 31.6 degrees to the beam direction, at 24.6 cm from the target, in order to be used as monitor, for normalization. The overall systematic uncertainty in normalization is estimated as ±6.0%. The typical irradiation time was 2 hours, during which the beam was multiscaled at 1 minute intervals. Then both the target and catcher foil were removed and placed in front of the X-ray detector within ∼ 5 minutes. Each catcher foil was used only once. For the detection of the delayed X-K rays, a germanium detector was used, with energy resolution of EFWHM = 600 eV in the energy range of the X-K rays. The lines to be analyzed were the Kα1 and Kα2 from Tb, Gd and Eu, from 40.9 keV to 44.5 keV. The energy separations of the peaks were typically around 700 eV, and therefore it was possible to separate the lines. Furthermore, as the relations between the X-Kα1 and X-Kα2 of each element are well known, they were used to check the consistency of the peak fit procedure. As we needed to separate isotopes with different half-lives producing the same X-rays, an automatic set of counting runs was programmed. Because of the half lives characteristics, the following set of counting periods for the delayed X-rays was used: 3 runs of 5 minutes, 3 runs of 15 minutes and 2 runs of 30 minutes each. For most of the energies, we have also accumulated some long runs lasting a few hours. More details about the fusion cross section measurements can be found in Refs. [5,6], including X-ray spectra. Elastic and inelastic angular distributions were measured using a set of seven surface barrier detectors at several bombarding energies. This set was placed 40 cm from the target, with 50 angular separations between two adjacent detectors, and had energy resolutions of the order of 350 keV. In front of each detector a set of collimators and circular apertures were used for definition of the solid angles and to eliminate slit-scattered particles. The relative solid angles of the detectors and the monitor were determined by Rutherford scattering from a thin 197 Au target. The elastic scattering could be resolved from the inelastic excitations of the first and second excited states of the target (E ∗ (2+ ) = 1.66 MeV and E ∗ (3− ) = 1.81 MeV). These are the most important inelastic target excitations. The projectile has no bound excited states. Fig. 1 shows a typical scattering spectrum. Although the statistics were quite poor for the inelastic excitations

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Fig. 2. (Color online.) Experimental excitation functions for several reaction channels. Note that σCF , σICF and σinel were measured directly while σTR and σDR were determined indirectly, using elastic scattering data.

of the first and second excited states of the target, it was possible to derive the sum of their integrated cross sections for most of the bombarding energies, by extrapolating the measured angular distribution range using the predictions of the FRESCO code [14], which was in good agreement with the data in the measured angular range. Total reaction cross sections (σTR ) were derived from elastic scattering data and cross sections for the total direct reactions (σTDR ) should be given by the sum of contributions from NCBU, direct α-transfer, one-neutron-transfer and inelastic scattering. That is, σTDR = σNCBU + σα -transfer + σ1n-transfer + σinel .

(1)

On the other hand, the capture of the α-particle following 9 Be breakup is a different process, which should contribute to σICF . However, our experiment cannot distinguish this process from direct α-transfer. Actually, analysis of particle-gamma coincidence measurements for the system 7 Li + 165 Ho [29], also with stable weakly bound projectile and a similar target mass, has shown that direct alpha transfer processes leading to the same exit channel of ICF are negligible. Assuming that the same situation occurs for our 9 Be + 144 Sm system, we include in σICF the contributions from all process leading to the formation of 148 Gd. In this way, another useful representation can be obtained by writing σDR = σNCBU + σ1n-transfer = σTR − σTF − σinel ,

(2)

where σDR is the direct reaction cross section, which is given by the combination of other measured cross section in the right-hand side of the above equation. σTF is the total fusion cross section, obtained by adding σCF and σICF . Fig. 2 shows the measured (σCF , σICF , σinel ) and derived (σTR and σDR ) cross sections. Interpolations of total reaction cross sections were used when the energies of elastic scattering and fusion (CF and ICF) did not match. One can observe that at sub-barrier energies most of σTR corresponds to σDR and that σinel is larger than σCF in this energy regime. At energies above the barrier CF is the dominant process. Table 1 shows the measured and derived cross sections.

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Table 1 Measured and derived cross sections. Ec.m. (MeV)

σCF (mb) [6]

σICF (mb) [6]

σTF (mb) [6]

σTR (mb)

28.2 29.7 30.1 31.1 32.0

1.41 ± 0.88

0.310 ± 0.21

1.72 ± 1.02

55.0 ± 5.2 111.9 ± 10.8

14.9 ± 5.4 44.6 ± 8.7 97.6 ± 9.5

1.66 ± 0.54 6.99 ± 3.17 13.99 ± 1.3

16.6 ± 6.1 51.6 ± 10.6 112 ± 14

34.8 35.8 37.5 38.6 39.5

295 ± 21 395 ± 27 496 ± 42 577 ± 39 669 ± 56

39.4 ± 6.7 54.1 ± 4.8 77.8 ± 4.7 85.4 ± 10.4 93.1 ± 10.0

335 ± 27 449 ± 24 574 ± 42 663 ± 40 762 ± 79

681.7 ± 6.1 [6] 784 ± 7 [6] 971 ± 9 [6] 1058 ± 9 [6]

41.4 45.2

770 ± 66

123 ± 8

893 ± 79

1292 ± 10 1422 ± 12

250.6 ± 2.2 [6] 354.4 ± 3.2 [6]

Table 2 Barrier parameters predicted by the SPP, where VB is the height, RB is the radius and h¯ ω is the curvature. 9 Be + 144 Sm

16 O + 144 Sm

RB (fm)

10.7

10.9

VB (MeV)

31.1

61.4

h¯ ω (MeV)

4.2

4.4

3. Comparison of σCF and σTF with theoretical predictions When one wants to discuss enhancement or hindrance of fusion cross sections in systems with low breakup threshold, one must have, as a starting point, a benchmark cross section not influenced by the binding energy of the system. If the adopted benchmark cross section is based on a theoretical model, the choice of the bare nuclear interaction is of crucial importance. Different choices of this interaction can lead to opposite conclusions about the influence of the breakup channel, going from enhancement to hindrance of fusion. We believe that the natural candidates to the bare nuclear interaction are double folding potentials based on realistic densities. In our calculations we adopt the São Paulo potential (SPP) [26–28], which uses the double folding model and contains no free parameter. In order to employ the SPP as a global potential, we used the reliable densities of Refs. [30,31]. The SPP has been successfully used to describe several reaction mechanisms for a large number of systems in a wide energy range [28,32,33], including weakly bound nuclei [34–36]. The barrier parameters predicted by the SPP for the 9 Be + 144 Sm and 16 O + 144 Sm systems are given in Table 2. We have performed coupled channel (CC) calculations using the FRESCO code [14]. We adopted the SPP as the bare nuclear interaction and included the first two excited states of 144 Sm in all calculations (β2 = 0.0881, r0 = 1.06 fm and λ = 2 for E ∗ = 1.66 MeV and β3 = 0.14, r0 = 1.06 fm and λ = 3 for E ∗ = 1.81 MeV). Fig. 3 shows the results of the theoretical predictions of two CC calculations, in comparison with the experimental TF and CF cross sections. Fig. 3a

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Fig. 3. (Color online.) Comparison of CF and TF data with predictions of two CC calculations. The dashed line is the result of a calculation taking into account the entrance channel and the two excited channels of 144 Sm with the lowest excitation energies. The solid line corresponds to results of a CC calculation including also resonant states in 9 Be. For details, see the text.

(left) shows the excitation functions in a logarithmic scale (suitable for sub-barrier energy region) and Fig. 3b (right) shows the same results in a linear scale (suitable for Ec.m. > VB ). The dashed line corresponds to the CC calculation including only the entrance channel and the above mentioned inelastic excitations of the 144 Sm target. In contrast with what has been done in previous analyses [5,6], couplings with 9 Be resonances were completely neglected. Comparing this curve with the CF data at above-barrier energies, we observe that the experimental cross section is suppressed by a factor of about 10%. On the other hand, the TF data in this energy range are quite close to the theoretical results. At sub-barrier energies the data are close to the theoretical predictions, except for the point at the lowest energy, where the experimental result is enhanced with respect to the theoretical one. The solid line represents the predictions of the second CC calculation. This calculation takes into account all the channels of the previous one but it includes also the main resonant states in the 9 Be projectile. That is, the resonances with E ∗ = 2.43 MeV, J π = 5/2− and E ∗ = 6.81 MeV, J π = 7/2− . They are members of the K = 3/2− band, with quadrupole deformation parameter β2 = 0.924. The E ∗ = 2.43 MeV resonance has a very short width (Γ = 0.77 keV) and therefore it can be treated as a bound state. Although the E ∗ = 6.81 MeV resonance has a large width of 1.2 MeV, the excitation energy of this state is quite large and it does not significantly affect the fusion cross section. This means that this calculation takes into account resonant breakup but not direct breakup. The inclusion of resonant states leads to a small reduction of the cross section at above-barrier energies and to a significant increase below the Coulomb barrier. In this way, the suppression of the CF data above the barrier is less pronounced and the TF data lies slightly above the theoretical curve. At sub-barrier energies, both the TF and CF data show a very small suppression with respect to this theoretical prediction. Investigating the influence of each of the two resonances on the fusion cross section, we found that all relevant effects came from the sharp resonance at E ∗ = 2.43 MeV. Considering that our bare nuclear interaction is based on realistic densities and that the relevant resonant states of 9 Be are explicitly included in this calculations, we conclude that the differences between the CF data and the solid curve arises from couplings with non-resonant continuum states (direct breakup).

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Fig. 4. (Color online.) Experimental fusion cross sections for the 16 O + 144 Sm and 9 Be + 144 Sm systems, reduced according to three different procedures. See text for details.

Fig. 5. (Color online.) Similar to Fig. 4, but here the cross sections were obtained from single-channel optical model calculations, using the SPP as the bare nuclear potential.

4. Comparing fusion data for weakly and tightly bound systems – The universal fusion function Another method to investigate the influence of the breakup channel on fusion of weakly bound systems is to compare the fusion data with those for a tightly bound projectile, on the same target. It is then convenient to represent the cross sections for the two systems in a single graphic. However, for a meaningful comparison, it is necessary to remove the differences arising from trivial static factors, like the system’s size and charges and the height and curvature of the Coulomb barrier. This can be done through the reduction of the cross section and/or collision energy. Several reduction prescription have been presented in the literature. Gomes et al. [37], suggested the transformations σF (3) σF → 1/3 1/3 (AP + AT )2 and 1/3

1/3

Ec.m. (AP + AT ) . (4) ZP ZT Eq. (3) takes care of removing the influence of the system sizes while Eq. (4) eliminates the dependence on the projectile’s and target’s charges. However, if one is interested in investigating Ec.m. →

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the influence of dynamic breakup effects (that is, breakup couplings) on the fusion cross section, it is necessary to use a reduction procedure that eliminates the effects of the weakly bound nucleons on the static Coulomb barrier. Therefore, the reduction procedure should be based on realistic values of VB and RB . These parameters can be evaluated from reliable bare interaction, as the SPP. A variety of reduction procedures have been proposed by several authors, like plotting σF vs. Ec.m. /VB [38–40], σF vs. E = Ec.m. − VB [41], and σF /πRB2 vs. Ec.m. /VB [2–4,42–44]. In Fig. 4 we compare experimental TF cross sections for two different systems: 9 Be + 144 Sm (weakly bound) and 16 O + 144 Sm (tightly bound). The results are reduced according to three of the above mentioned methods. The barrier parameters used in the reductions in panels (b) and (c) are those shown in Table 2. We see that with the reductions based on the realistic barrier parameters ((b) and (c)) the curves for the weakly and tightly bound systems are much closer. This could suggest that the remaining differences were due to dynamic effects of channel coupling, with important contribution from the breakup channel. However, this assumption would be very wrong. There is still a strong contribution from static effects, arising from the potential barriers. To illustrate this point, we show a similar plot in Fig. 5, where the fusion cross sections were obtained from single-channel optical model calculations, using the SPP as the bare nuclear interaction. Of course, these results are not influenced by any kind of channel coupling. We see that the cross sections reduced by any of the three procedures remains quite different. Therefore, residual static effects survived after the reduction. In fact, the differences in the curves for the two systems are still more pronounced than in the previous figure, which indicates that they were attenuated by the couplings with bound and breakup channels. A better elimination of static effects associated with low breakup threshold can be achieved through a recently proposed reduction method [16,17]. This method is based on realistic values of the radius and height of the Coulomb barrier (RB and VB ), as the methods used in panels (b) and (c) of Figs. 4 and 5, but it takes into account also the barrier curvature, h¯ ω. The fusion cross section and the collision energy are reduced to dimensionless variables, according to the transformations Ec.m. − VB 2Ec.m. σF and x = . (5) F (x) = 2 h¯ ω hωR ¯ B In Refs. [16,17], F (x) was called fusion function. The reduction method of Eq. (5) is based on Wong’s approximation [45] for the fusion cross section,    2π(E − VB ) ¯ω W 2 h ln 1 + exp . (6) σ F = RB 2E h¯ ω If the fusion cross section can be approximated by Eq. (6), the fusion function takes the simple form   (7) F0 (x) = ln 1 + exp(2πx) . It is clear that F0 (x) is independent of the system. Owing to this feature, F0 (x) was called [16] the universal fusion function (UFF). The UFF can then be used as a benchmark to assess the influence of different couplings on fusion cross sections. In this way, reduced fusion data for several systems can be compared with F0 (x) in a single plot. For this purpose, we evaluate experimental fusion functions, Fexp (x), using experimental fusion cross sections in Eq. (5). In Fig. 6, we illustrate the elimination of the static effects achieved by the reduction method of Eq. (5). As in Fig. 5, we show reduced fusion cross sections obtained from optical model

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Fig. 6. (Color online.) Fusion functions for the 9 Be + 144 Sm and 16 O + 144 Sm systems. See text for details.

calculations for the 16 O + 144 Sm and 9 Be + 144 Sm systems. In panels (a) and (b) the reduced cross sections for each of these systems is compared to the UFF. In both cases the reduced cross sections are very close to the UFF, and consequently they are close to each others. This indicates that this reduction method leaves very weak residual static effects. In fact, the pronounced deviations observed for 9 Be + 144 Sm at sub-barrier energies arises from the inaccuracy of the Wong’s approximation. This limitation can be corrected, as discussed below. Despite the advantages of the reduction method of Eq. (5), it has two shortcomings. The first is that Wong’s approximation is inaccurate at sub-barrier energies for light systems, where ZP ZT < 500 [16,17]. The second is that the difference between Fexp (x) and the UFF should be attributed to the combined effect of couplings with all bound and unbound channels. In this way, the effects of couplings with all channels are entangled. This is an undesirable feature, since we are interested exclusively in the influence of the breakup channel. These limitations of the method can be avoided through the replacement of Fexp (x) by the modified experimental fusion function [16,17], Fexp (x) → F¯exp (x) = Fexp (x) ×

F0 (x) . FCC (x)

(8)

In this equation, FCC (x) corresponds to the fusion cross section of a proper CC calculation (including the effects of all relevant bound channels), reduced by the prescription of Eq. (5). The modified experimental fusion function of Eq. (8) has an important property. In a collision of tightly bound nuclei where couplings with all relevant bound channels are correctly account for, it reduces to the UFF. Therefore, the differences between F¯exp (x) and the UFF can be traced back to breakup coupling. We used the method of fusion functions to analyze the same experimental CF and TF cross sections of Fig. 3. For comparison, we included also the F¯exp (x) for the tightly bound 16 O + 144 Sm system. The plots are shown both in linear (a) and logarithmic (b) scales. The CC calculation used to evaluate FCC (x) took into account only the excited states of the target, as for the dashed line in Fig. 3. Resonances in 9 Be were left out. In this way, the differences between F¯exp (x) and the UFF can be attributed to couplings with both resonant and non-resonant breakup states. First, we observe that for the 16 O + 144 Sm system F¯exp (x) coincides with the UFF, in all energy regions of the plot. (See Fig. 7.) The reduced TF cross section for 9 Be + 144 Sm is also close to the UFF for all collision energies. On the other hand, the reduced CF cross section for

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Fig. 7. (Color online.) F¯exp for 16 O + 144 Sm and 9 Be + 144Sm (for details see the text).

Fig. 8. (Color online.) Energy dependences of σDR , σNCBU , σICF+α -transfer and σ1n-transfer . For details see the text.

this system is very close to the UFF at sub-barrier energies but it falls slightly below the UFF at energies above the barrier. These results are consistent with those in Section 3. 5. Derivation of non-capture breakup cross sections When experiments with 9 Be beams involve the measurement of alpha particles in coincidence, the yield of alpha particles corresponds to σα –α coincidences = σNCBU + σ1n-transfer .

(9)

Comparing Eqs. (2) and (9), one observes that the present data and procedure to determine σDR give the same information as when one performs α–α coincidence experiments. In order to estimate the non-capture break-up cross section (σNCBU ), we have performed coupled reaction channel calculations using the FRESCO code to estimate the one-neutron transfer cross sections. The cross section σNCBU was then obtained by the difference between the measured cross sections σRF and σTF , and the calculated 1n transfer cross section, according to

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Eq. (2). In Ref. [5] the ground and the first excited states of 145 Sm were included in the calculations and the spectroscopic factors were assumed to be equal to one, in both cases. In the present work we want to improve these calculations. The ground state Q-value (Qg.s. ) for this transfer reaction is 5.092 MeV. As the transfered particle has no charge the optimal energy of the residual nucleus 145 Sm should be equal to Qg.s. [46]. Since this excitation energy is very high, the final state is in the quasi-continuum part of the spectrum, where the density of states is very high. In this way, the transfer reaction populates several states with excitation energies around the optimal value, with very different nuclear structure properties. For simplicity, we assume that the strength for this set of states can be approximated by that for the transition to a Harmonic − Oscillator eigen-state, with excitation energy close to the optimal value. We used a 32 state with excitation energy of 5.09 MeV in 145 Sm. As we are not projecting this state on actual states of 145 Sm, there is no spectroscopic factor involved. In the finite-range transfer calculation, the prior interaction was used, and the standard Wood Saxon parameters of the 8 Be + n and 144 Sm + n interactions were varied in order to obtain the experimental separation energies. The results for the one-neutron transfer cross sections were very similar to the ones obtained in Ref. [5]. Fig. 8 shows the derived non-capture breakup cross section σNCBU , the calculated σ1n-transfer and the measured sum of σα -transfer + σICF . The direct reaction cross section is essentially the same as the non-capture breakup cross section. 6. Investigation of the energy dependence of the optical potential through analysis of elastic scattering angular distributions The threshold anomaly (TA) is a well known phenomenon involving the imaginary and the real parts of the nuclear interaction, as functions of the collision energy. It consists of a sharp decrease of the imaginary part as decreasing energy [22,23], together with a localized bell shape energy dependence of the real part. These behaviors occur in the vicinity of the Coulomb barrier. The real and imaginary parts of the potential are interrelated through a dispersion relation [23,47]. The decrease of the imaginary potential occurs because most reaction channels close at very low energies. The increase of the real potential corresponds to an additional attraction, which leads to the enhancement of the fusion cross section at sub-barrier energies. The origin of this effect is the attractive polarization potential generated by the couplings with inelastic and transfer channels. However, couplings with the breakup channel involves continuum states and lead to repulsive polarization potentials [48]. The net effect of couplings with bound and continuum states may lead to attractive or to repulsive polarization potentials, depending on the details of the couplings. Therefore, in collisions of weakly bound systems, TA can be observed or not. In some cases, the behavior of the imaginary potential can even show the opposite behavior, increasing as the collision energy approaches the height of the Coulomb barrier. This phenomenon was called breakup threshold anomaly (BTA) [24,25]. The elastic scattering of weakly bound systems at near-barrier energies has been studied by several authors and the energy dependences of the real and imaginary parts of the potentials were investigated. These studies did not show a systematic trend. The conclusions were strongly dependent on the characteristics of the weakly bound projectile and also on the target nucleus. BTA has been observed in elastic scattering of 6 He and 6 Li projectiles [24,49–55]. However, in collisions of 7 Li projectiles, the situation is quite different. The analysis of the elastic scattering data lead to the usual TA or to no anomaly at all, depending on the target nucleus [51,53,56,57]. It has been shown [58] that the coupling of the bound excited state of 7 Li is responsible for the vanishing of the usual TA for some systems.

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Studies of elastic scattering of 9 Be projectiles with different targets have also been performed. The resulting energy behaviors of the real and imaginary parts of the potential showed contradictory trends [25,59–61]. Previous elastic scattering experiments with the 9 Be + 144 Sm system [5,6] could not be used to investigate the existence of such anomalies because there were no sub-barrier energy data. In this paper we complement the results of the previous ones, reporting experimental angular distributions at two sub-barrier energies (Elab = 30 and 31.5 MeV), and two above the barrier (Elab = 44 and 48 MeV). At sub-barrier energies, the collision is dominated by the Coulomb potential. Therefore, information about the nuclear potential can only be extracted from the data at very large angles, since they correspond to frontal collisions. For this reason, we performed measurements at angles up to θlab = 171◦ . In order to reach conclusive results about the anomalies of the optical potential, it is also necessary to use small angular step, especially in the rainbow region. We adopted the angular step θ = 2◦ . Different approaches can be used to extract the energy dependence of the interaction potential from elastic scattering data. One widely used is to parametrize the real and imaginary parts of the nuclear potential by Woods–Saxon functions and fit the parameters as to reproduce the elastic scattering data. The ambiguities can be reduced if one fixes the value of the reduced radius and fit only the strength and the diffusivity parameters. The study of the energy dependences is usually based on the values of the potentials at the sensitivity radius, where the potentials producing good fits of the angular distribution cross. This procedure also reduces the ambiguities of the fits. A second frequently used approach is to work with renormalized double-folding potentials. In this case, the real part of the potential is given by the double-folding model, using realistic densities. The same r-dependence is assumed for the imaginary part of the potential. However, these potentials are multiplied by the energy-dependent renormalization factors NR (E) and NI (E), which are fitted to the elastic scattering data. Threshold anomalies are then investigated through the energy dependence of these renormalization factors. A third approach is to use a double-folding potential for the real part and to parametrize the imaginary part by a Woods– Saxon function. The renormalization factor, NR (E), and the Woods–Saxon parameters are then fit to the elastic scattering data. These three approaches have been compared in Refs. [52,57,62], for collisions of the stable weakly bound projectiles 6,7 Li with a 27 Al target. The results were similar. In this work we used the double-folding SPP for the real part of the nuclear interaction and assumed the same r-dependence for the imaginary part. At energies very close to the Coulomb barrier, where coupling effects are important, one needs to use two free parameters in the SPP, the normalization of the real and imaginary parts of the potential, NR (E) and NI (E). Fig. 9 shows the fits of the elastic scattering data varying the strengths of the real and imaginary potentials. One can see that very good fits were obtained for all the studied energies. In Fig. 10 we investigate the energy dependences of the real and imaginary parts of the nuclear interaction. The figure shows the renormalization factors NR (E) and NI (E) which best fits the elastic angular distributions. The uncertainties in these quantities were obtained by varying the parameters up to the point in which the total χ 2 value of the fit increases in one unity, corresponding approximately to a confidence level of 68.3%. From Fig. 10 we conclude that neither the usual TA nor BTA are observed for this system. So, the effect of the repulsive polarization potential arising from breakup coupling compensates the attractive polarization potential due to inelastic target excitations. In this way, the real and imaginary parts of the nuclear interaction are nearly energy-independent. This behavior is similar to the one observed for the 7 Li + 27 Al system [63]. In this case, the breakup polarization potential was calculated. It was shown to be repulsive, compensating the attractive effect of the inelastic couplings. This fact is also in agree-

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Fig. 9. (Color online.) Elastic angular distributions for the studied energies. The full lines represent the best fits varying the strength of the potentials. For details see the text.

Fig. 10. (Color online.) Energy dependence of the real (NR ) and imaginary parts (NI ) of the SPP.

ment with the behavior of fusion cross section studied in Sections 3 and 4. That is, no strong enhancement or suppression is observed at energies near the Coulomb barrier for this system. 7. Simultaneous optical model calculations of elastic scattering and fusion Another approach to investigate the effect of the breakup on elastic scattering and fusion of weakly bound systems is to perform a simultaneous optical model analysis of the cross sections for these two processes. Udagawa et al. [18–20] proposed an analysis along these lines, splitting the imaginary potential into volumetric and superficial contributions. This procedure is quite reasonable since the imaginary potential arises from several processes, with very different natures. The volumetric part is associated with complete fusion while the surface term accounts for the attenuation of the incident current owing to direct reactions, which takes place at larger values of the radial distance. In this way, we can write W (r) = WF (E, r) + WDR (E, r).

(10)

Above, we used a notation which emphasizes the fact that the imaginary potential is energydependent. In collisions of weakly bound nuclei, the breakup channel gives an important con-

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Fig. 11. (Color online.) Energy dependence of the volume (WF ) and surface (WDR ) terms of the imaginary potentials at the sensitivity radius.

tribution to the potential WDR (E, r). The real part of the potential should also have polarization corrections, associated with these imaginary terms through dispersion relations. In our analysis, we parametrize WF (E, r) and WDR (E, r) by a Woods–Saxon function and its derivative, respectively, in the forms WF (r, E) = WF (E) × f (r)

(11)

and WDR (r, E) = WDR (E) × 4aDR

df (r) dr

(12)

with f (xi ) =

1 , 1 + exp(xi ) 1/3

xi =

r − Ri , ai

i = F, DR.

(13)

1/3

Above, Ri = r0i (AP + AT ), with r0i and ai standing for the reduced radius and the diffuseness parameters. The Woods–Saxon parameters of the optical potentials are extracted by simultaneous fits of CF and elastic scattering data. We then look for threshold anomalies in the energy-dependences of WF (r, E) and WDR (r, E). To obtain the real part of the polarization potentials corresponding to WF (r, E) and WDR (r, E), we use the dispersion relation and assume that they have the same radial dependence as their imaginary counterpart. In our study, we evaluate all potentials at the sensitivity radius, which for our system has the average value Rs = 11.8 fm. The total reaction cross section is calculated by using the full absorption potential W as σR (E) =

(+)

2  (+) χ0 W (E) χ0 , h¯ v

(14)

(+)

where χ0 is the distorted wave in the elastic channel, with scattering boundary condition. The fusion and direct reaction cross sections are given by (+)

k  (+) χ0 WF (E) χ0 , E (+)

k  (+) χ0 WDR (E) χ0 . σDR (E) = E

σF (E) =

(15) (16)

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Fig. 12. (Color online.) Radial dependence of the volume (solid line) and surface (dashed line) term of the imaginary potentials. Panels (a) and (b) correspond to results for different choices of the Wood–Saxon parameters. For details see the text.

Refs. [64,65] reported the energy-dependences of WF (E) and WDR (E) for the 9 Be + 144 Sm system, extracted from elastic angular distributions measured at six above-barrier energies. In the present work, we perform a similar study including new data. We considered elastic angular distributions at ten collision energies, being two below the barrier. In Fig. 11 we show the behavior of the imaginary parts of the volume and surface terms of the optical potential, evaluated at the mean sensitivity radius, Rs = 11.8 fm. One can see that the volume part (WF ), expressing the effects of complete fusion, shows the well known threshold anomaly. On the other hand, the surface part (WDR ), which is mainly influenced by breakup coupling, shows a behavior similar to the BTA. In the following we discuss in details the fact that among several families of potentials which fit simultaneously fusion and elastic scattering data, with χ 2 values smaller than 1.2, the physically acceptable parameters of the optical potential are those which satisfy the condition that the strength of the fusion imaginary potential WF must be smaller than the direct imaginary potential WDR at the tail region of the potentials. This assumption is supported by previous works [66–68] which show that for weakly bound systems, |WF | < |WDR | at the strong absorption radius for energies around the Coulomb barrier. This constraint may tell us the reasonable values for rDR , and consequently which is the region where breakup starts to occur. Fig. 12 shows the radial dependences of the volumetric (full curve), WF , and surface (dashed curve), WDR , imaginary potentials. The results are for Elab = 33 MeV, and they were obtained with two choices of the Wood–Saxon parameters. In (a) we used r0F = 1.1 fm, aF = 0.44 fm, r0DR = 1.2 fm and a0DR = 1.0 fm. One can observe that although |WF | < |WDR | at the tail region, this solution is not satisfactory, since most of the surface potential is inside the fusion volume potential. In Fig. 12b, we show the results for the parameters r0F = 1.2 fm, a0F = 0.65 fm, r0DR = 1.6 fm and aDR = 0.64 fm. This second choice of parameter is much more reasonable, since their values are similar to ones usually adopted in the literature. Actually, if one fixes r0F = 1.2 fm and varies r0DR , the χ 2 values for the simultaneous fits of fusion and elastic scattering decreases as r0DR increases. In this fit procedure the diffuseness parameters and the depth of the potentials were varied. The results are shown in Fig. 13 for the energy Elab = 41 MeV. Similar results were obtained for the other energies. In Fig. 13a the r0DR value is equal to 1.1 fm, in Fig. 13b it is 1.2 fm and so on, until Fig. 13f, where r0DR = 1.6 fm. If r0F is fixed at 1.1 fm or 1.3 fm, the acceptable values for r0DR moves from 1.5 fm to 1.7 fm, respectively. So, we con-

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Fig. 13. (Color online.) Radial dependence of volume (solid line) and the surface (dashed line) terms of the imaginary potential, for different values of the parameters r0F and r0DR . For details see the text.

clude that the direct reactions, which for the present system is essentially the breakup process, might start to occur at a reduced distance of the order of 1.6 fm. 8. Conclusions We have performed a comprehensive study of the cross sections for several processes taking place in 9 Be + 144 Sm collisions. Our study was based on results of direct measurements available in the literature, on new experimental results reported in the present paper, and on indirect determinations of cross sections. We have discussed the available CF, ICF, elastic and inelastic data and derived the NCBU cross section trough an indirect procedure, using elastic scattering data and an improved theoretical estimate of the one-neutron transfer cross section. This method is an alternative to experiments using α–α coincidence measurements. We found that at sub-barrier energies the predominant process is NCBU, while CF predominates at energies above the barrier. We used different approaches to analyze the data, and all of them show consistent results. Suppression of the order of 10% on the CF cross section occurs at energies above the barrier, as compared with predictions from a realistic double folding potential or with the similar tightly bound 16 O + 144 Sm system, whereas the TF cross section agrees with the predictions. At sub-barrier energies there is agreement with coupled channel calculations that do not take into account neither prompt nor sequential breakups, while if the sequential breakup is considered, there is a small suppression owing to the prompt breakup not taken into account in the calculations. We have investigated the energy dependences of the real and imaginary parts of the nuclear interaction, obtained by fitting elastic scattering data. Contrasting with the TA, observed in col-

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lisions of tightly bound systems, and the BTA, observed for some weakly bound systems, the real and imaginary potentials resulting from our optical model analysis were shown to be nearly energy-independent. We argued that this result is an indication that the repulsion due to breakup coupling cancels the attraction arising from couplings with bound channels. We have shown that different conclusions are reached if the imaginary potential is split into surface and volume terms, and each of these potentials are derived from simultaneous analysis of elastic scattering and fusion data. In this case, the volume term associated with CF exhibits the usual TA whereas the surface one, dominated by breakup coupling, shows BTA. Finally, also from the simultaneous analysis of fusion and elastic scattering data and from physical consideration constraints imposed to the imaginary potentials, we were able to evaluate the distance where breakup might start to occur, before reaching the Coulomb barrier radius. Acknowledgements This work was supported in part by the FAPERJ, CNPq, the PRONEX and the Prosul. We gratefully acknowledge discussions with Dr. A.J. Pacheco for his suggestions to improve the definitive version of this manuscript. References [1] L.F. Canto, P.R.S. Gomes, R. Donangelo, M.S. Hussein, Phys. Rep. 424 (2006) 1. [2] M. Dasgupta, P.R.S. Gomes, D.J. Hinde, S.B. Moraes, R.M. Anjos, A.C. Berriman, R.D. Butt, N. Carlin, J. Lubian, C.R. Morton, J.O. Newton, A. Szanto de Toledo, Phys. Rev. C 70 (2004) 024606. [3] M. Dasgupta, D.J. Hinde, R.D. Butt, R.M. Anjos, A.C. Berriman, N. Carlin, P.R.S. Gomes, C.R. Morton, J.O. Newton, A. Szanto de Toledo, K. Hagino, Phys. Rev. Lett. 82 (1999) 1395. [4] M. Dasgupta, D.J. Hinde, K. Hagino, S.B. Moraes, P.R.S. Gomes, R.M. Anjos, R.D. Butt, A.C. Berriman, N. Carlin, C.R. Morton, J.O. Newton, A. Szanto de Toledo, Phys. Rev. C 66 (2002) 041602(R). [5] P.R.S. Gomes, I. Padrón, E. Crema, O.A. Capurro, J.O. Fernández Niello, G.V. Martí, A. Arazi, M. Trotta, J. Lubian, M.E. Ortega, A.J. Pacheco, M.D. Rodriguez, J.E. Testoni, R.M. Anjos, L.C. Chamon, M. Dasgupta, D.J. Hinde, K. Hagino, Phys. Lett. B 634 (2006) 356. [6] P.R.S. Gomes, I. Padrón, E. Crema, O.A. Capurro, J.O. Fernández Niello, A. Arazi, G.V. Martí, J. Lubian, M. Trott, A.J. Pacheco, J.E. Testoni, M.D. Rodriguez, M.E. Ortega, L.C. Chamon, R.M. Anjos, R. Veiga, M. Dasgupta, D.J. Hinde, K. Hagino, Phys. Rev. C 73 (2006) 064606. [7] P.R.S. Gomes, I. Padrón, A.O. Capurro, J.O. Fernández Niello, G.V. Marti, A.J. Pacheco, A. Arazi, J. Lubian, E. Crema, X-Ray Spectrom. 37 (2008) 512. [8] R.G. Stokstad, Y. Eisen, S. Kaplanis, D. Pelte, U. Smilansky, I. Tserruya, Phys. Rev. Lett. 41 (1978) 465. [9] D.E. Di Gregorio, A.J. Pacheco, J.O. Fernández Niello, D. Abriola, S. Gil, A.O. Machiavelli, J. Testoni, P.R. Pascholati, V.R. Vanin, N. Carlin Filho, M. Coimbra, R. Liguori Neto, P.R.S. Gomes, G.R. Stokstad, Phys. Lett. B 176 (1986) 322. [10] D.E. Di Gregorio, M. di Tada, D. Abriola, M. Elgue, A. Etchegoyen, M.C. Etchegoyen, J.O. Fernández Niello, A.M.J. Ferrer, S. Gil, A.O. Macchiavelli, A.J. Pacheco, J.E. Testoni, P.R.S. Gomes, V.R. Vanin, R. Liguori Neto, E. Crema, R.G. Stokstad, Phys. Rev. C 39 (1989) 516. [11] P.R.S. Gomes, I.C. Charret, R. Wanis, G.M. Sigaud, V.R. Vanin, R. Liguori Neto, D. Abriola, O.A. Capurro, D.E. Di Gregorio, M. di Tada, G. Duchene, M. Elgue, A. Etchegoyen, J.O. Fernández Niello, A.M.J. Ferrero, S. Gil, A.O. Macchiavelli, A.J. Pacheco, J.E. Testoni, Phys. Rev. C 49 (1994) 245. [12] A. Di Pietro, P. Figuera, F. Amorini, C. Angulo, G. Cardella, S. Cherubini, T. Davinson, D. Leanza, J. Lu, H. Mahmud, M. Milin, A. Musumarra, A. Ninane, M. Papa, M.G. Pellegriti, R. Raabe, F. Rizzo, C. Ruiz, A.C. Shotter, N. Soic, S. Tudisco, L. Weissman, Phys. Rev. C 69 (2004) 044613. [13] A. Diaz-Torres, I.J. Thompson, Phys. Rev. C 65 (2002) 024606. [14] I.J. Thompson, Comput. Phys. Rep. 7 (1988) 167. [15] C. Signorini, Z.H. Liu, Z.C. Li, K.E.G. Löbner, L. Müller, M. Ruan, K. Rudolph, F. Soramel, C. Zotti, A. Anddrighetto, L. Stroe, A. Vitturi, H.Q. Zhang, Eur. Phys. J. A 5 (1999) 7.

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